Reproducing kernel Hilbert space method is utilized in this paper as an efficient approach to solve singular fourth order boundary value problems of mixed form Fredholm-Volterra integro-differential equations. In order to obtain the required nodal values, the algorithm developed using two smooth reproducing kernel functions. The solution philosophy is based on applying the Gram-Schmidt process to the kernel function obtained in the space W 2 5 [ 0 , 1 ] to produce an orthogonal basis. From that point onward, the orthogonal basis was built for the purpose of formulating and utilizing numerical solutions in the same space. Some linear and nonlinear numerical issues were broken down to delineate the technique and affirm the execution of the proposed strategy. The numerical outcomes emphasize the method role in improving the initial approximation, handling boundary conditions, and refining the solution throughout the iterative process to shed light on such singular equations.
KeywordsHilbert SpaceIntegro-Differential EquationGram-Schmidt ProcessFredholm-Volterra Integral EquationsReproducing Kernel Function
Kanwal, R.P. (1997) Linear Integral Equations Theory and Technique.
Wazwaz, A. (2006) A Comparison Study between the Modified Decomposition Method and the Traditional Methods for Solving Nonlinear Integral Equations. Applied Mathematics and Computation , 181, 1703-1712. https://doi.org/10.1016/j.amc.2006.03.023
Bloom, F. (1980) Asymptotic Bounds for Solutions to a System of Damped Integrodifferential Equations of Electromagnetic Theory. Journal of Mathematical Analysis and Applications , 73, 524-542. https://doi.org/10.1016/0022-247x(80)90297-8
Holmåker, K. (1993) Global Asymptotic Stability for a Stationary Solution of a System of Integro-Differential Equations Describing the Formation of Liver Zones. SIAM Journal on Mathematical Analysis , 24, 116-128. https://doi.org/10.1137/0524008
Doddrell, D.M., Forbes, L.K. and Crozier, S. (1997) Caluculating Current Densities and Fields Produced by Shielded Magnetic Resonance Imaging Probes. SIAM Journal on Applied Mathematics , 57, 401-425. https://doi.org/10.1137/s0036139995283110
Zaremba, S. (1907) L’equation biharminique et une class remarquable defonctions foundamentals harmoniques. Bulletin International de l ’ Academie des Sciences de Cracovie , 147-196. https://www.biodiversitylibrary.org/item/276656?utm_source=chatgpt.com#page/11/mode/1up
Mercer, J. (1909) XVI. Functions of Positive and Negative Type, and Their Connection the Theory of Integral Equations. Philosophical Transactions of the Royal Society of London. Series A , Containing Papers of a Mathematical or Physical Character , 209, 415-446. https://doi.org/10.1098/rsta.1909.0016
Aronszajn, N. (1950) Theory of Reproducing Kernels. Transactions of the American Mathematical Society , 68, 337-404. https://doi.org/10.1090/s0002-9947-1950-0051437-7
Cui, M. and Lin, Y. (2009) Nonlinear Numerical Analysis in the Reproducing Kernel Space. Nova Science.
Berlinet, A. and Agnan, C.T. (2004) Reproducing Kernel Hilbert Space in Probability and Statistics. Kluwer Academic Publishers. https://doi.org/10.1007/978-1-4419-9096-9
Daniel, A. (2003) Reproducing Kernel Spaces and Applications. Springer.
Lin, Y.Z., Cui, M.G. and Yang, L.H. (2006) Representation of the Exact Solution for a Kind of Nonlinear Partial Differential Equations. Applied Mathematics Letters , 19, 808-813.
Wang, W., Han, B. and Yamamoto, M. (2013) Inverse Heat Problem of Determining Time-Dependent Source Parameter in Reproducing Kernel Space. Nonlinear Analysis : Real World Applications , 14, 875-887. https://doi.org/10.1016/j.nonrwa.2012.08.009
Jiang, W. and Lin, Y. (2010) Approximate Solution of the Fractional Advection-Dispersion Equation. Computer Physics Communications , 181, 557-561. https://doi.org/10.1016/j.cpc.2009.11.004
Jiang, W. and Chen, Z. (2014) A Collocation Method Based on Reproducing Kernel for a Modified Anomalous Subdiffusion Equation. Numerical Methods for Partial Differential Equations , 30, 289-300. https://doi.org/10.1002/num.21809
Arqub, O.A. (2016) Approximate Solutions of Dass with Nonclassical Boundary Conditions Using Novel Reproducing Kernel Algorithm. Fundamenta Informaticae , 146, 231-254. https://doi.org/10.3233/fi-2016-1384
Jiang, W. and Chen, Z. (2013) Solving a System of Linear Volterra Integral Equations Using the New Reproducing Kernel Method. Applied Mathematics and Computation , 219, 10225-10230. https://doi.org/10.1016/j.amc.2013.03.123
Arqub, O.A., Al-Smadi, M. and Shawagfeh, N. (2013) Solving Fredholm Integro-Differential Equations Using Reproducing Kernel Hilbert Space Method. Applied Mathematics and Computation , 219, 8938-8948. https://doi.org/10.1016/j.amc.2013.03.006
Abu Arqub, O. and Al-Smadi, M. (2014) Numerical Algorithm for Solving Two-Point, Second-Order Periodic Boundary Value Problems for Mixed Integro-Differential Equations. Applied Mathematics and Computation , 243, 911-922. https://doi.org/10.1016/j.amc.2014.06.063
Geng, F.Z. and Qian, S.P. (2013) Reproducing Kernel Method for Singularly Perturbed Turning Point Problems Having Twin Boundary Layers. Applied Mathematics Letters , 26, 998-1004. https://doi.org/10.1016/j.aml.2013.05.006
Geng, F.Z., Qian, S.P. and Li, S. (2014) A Numerical Method for Singularly Perturbed Turning Point Problems with an Interior Layer. Journal of Computational and Applied Mathematics , 255, 97-105. https://doi.org/10.1016/j.cam.2013.04.040
Geng, F.Z. and Qian, S.P. (2015) Modified Reproducing Kernel Method for Singularly Perturbed Boundary Value Problems with a Delay. Applied Mathematical Modelling , 39, 5592-5597. https://doi.org/10.1016/j.apm.2015.01.021
Wazwaz, A. (2001) A Reliable Algorithm for Solving Boundary Value Problems for Higher-Order Integro-Differential Equations. Applied Mathematics and Computation , 118, 327-342. https://doi.org/10.1016/s0096-3003(99)00225-8
Arikoglu, A. and Ozkol, I. (2005) Solution of Boundary Value Problems for Integro-Differential Equations by Using Differential Transform Method. Applied Mathematics and Computation , 168, 1145-1158. https://doi.org/10.1016/j.amc.2004.10.009
Yıldırım, A. (2008) Solution of BVPs for Fourth-Order Integro-Differential Equations by Using Homotopy Perturbation Method. Computers & Mathematics with Applications , 56, 3175-3180. https://doi.org/10.1016/j.camwa.2008.07.020
Hashim, I. (2006) Adomian Decomposition Method for Solving BVPs for Fourth-Order Integro-Differential Equations. Journal of Computational and Applied Mathematics , 193, 658-664. https://doi.org/10.1016/j.cam.2005.05.034
Wazwaz, A.M. (2001) Linear and Nonlinear Integral Equations Methods and Applications. Springer.
Çağlar, H., Çağlar, N. and Özer, M. (2009) B-Spline Solution of Non-Linear Singular Boundary Value Problems Arising in Physiology. Chaos , Solitons & Fractals , 39, 1232-1237. https://doi.org/10.1016/j.chaos.2007.06.007
Hiltmann, P. and Lory, P. (1983) On Oxygen Diffusion in a Spherical Cell with Michaelis-Menten Oxygen Uptake Kinetics. Bulletin of Mathematical Biology , 45, 661-664. https://doi.org/10.1016/s0092-8240(83)80019-6